Introducton to crcut analyss OUTLINE Electrcal quanttes Charge Current Voltage Power The deal basc crcut element Sgn conventons Current versus voltage (I-V) graph Readng: 1.2, 1.3,1.6 Lecture 2, Slde 1 Classfcaton of Materals Solds n whch all electrons are tghtly bound to atoms are nsulators. Examples: Solds n whch the outermost atomc electrons are free to move around are metals. Metals typcally have ~1 free electron per atom Examples: Electrons n semconductors are not tghtly bound and can be easly promoted to a free state. Examples: Lecture 2, Slde 2 1
Lecture 2, Slde 3 Electrc Charge Electrcal effects are due to separaton of charge electrc force (voltage) charges n moton electrc flow (current) Macroscopcally, most matter s electrcally neutral most of the tme. Exceptons: clouds n a thunderstorm, people on carpets n dry weather, plates of a charged capactor, etc. Mcroscopcally, matter s full of electrc charges Electrc charge exsts n dscrete quanttes, ntegral multples of the electronc charge -1.6 x 10-19 Coulomb Lecture 2, Slde 4 2
Electrc Current Defnton: rate of postve charge flow Symbol: Unts: Coulombs per second Amperes (A) Note: Current has polarty. = dq/dt where q = charge (Coulombs) t = tme (n seconds) Lecture 2, Slde 5 André-Mare Ampère's 1775-1836 Electrc Current Examples 1. 10 5 postvely charged partcles (each wth charge 1.6 10-19 C) flow to the rght (x drecton) every nanosecond I 5 19 Q 10 1.6 10 5 = = = 1.6 10 A 9 t 10 2. 10 5 electrons flow to the rght (x drecton) every mcrosecond 5 19 Q 10 1.6 10 I = = = 1.6 10 9 t 10 5 A Lecture 2, Slde 6 3
Current Densty Defnton: rate of postve charge flow per unt area Symbol: J Unts: A / cm 2 Example 1: Wre attached to end C1 2 cm 1 cm 10 cm Suppose we force a current of 1 A to flow from C1 to C2: Electron flow s n -x drecton: C / sec = 6.25 10 19 1.6 10 C / electron 1 18 X Semconductor wth 10 18 free electrons per cm 3 C2 electrons sec Lecture 2, Slde 7 Current Densty Example (cont d) Example 2: Typcal dmensons of ntegrated crcut components are n the range of 1 µm. What s the current densty n a wre wth 1 µm² area carryng 5 ma? Lecture 2, Slde 8 4
Electrc Potental (Voltage) Defnton: energy per unt charge Symbol: v Unts: Joules/Coulomb Volts (V) v = dw/dq Alessandro Volta (1745 1827) where w = energy (n Joules), q = charge (n Coulombs) Note: Potental s always referenced to some pont. a b Subscrpt conventon: v ab means the potental at a mnus the potental at b. v ab v a -v b Lecture 2, Slde 9 Electrc Power Defnton: transfer of energy per unt tme Symbol: p Unts: Joules per second Watts (W) Concept: p = dw/dt = (dw/dq)(dq/dt) = v As a postve charge q moves through a drop n voltage v, t loses energy energy change = qv rate s proportonal to # charges/sec James Watt 1736-1819 Lecture 2, Slde 10 5
The Ideal Basc Crcut Element v _ Polarty reference for voltage can be ndcated by plus and mnus sgns Reference drecton for the current s ndcated by an arrow Attrbutes: Two termnals (ponts of connecton) Mathematcally descrbed n terms of current and/or voltage Cannot be subdvded nto other elements Lecture 2, Slde 11 A Note about Reference Drectons A problem lke Fnd the current or Fnd the voltage s always accompaned by a defnton of the drecton: - v In ths case, f the current turns out to be 1 ma flowng to the left, we would say = -1 ma. In order to perform crcut analyss to determne the voltages and currents n an electrc crcut, you need to specfy reference drectons. There s no need to guess the reference drecton so that the answers come out postve. Lecture 2, Slde 12 6
Sgn Conventon Example Suppose you have an unlabelled battery and you measure ts voltage wth a dgtal voltmeter (DVM). It wll tell you the magntude and sgn of the voltage. a b 1.401 DVM Wth ths crcut, you are measurng v ab. The DVM ndcates 1.401, so v a s lower than v b by 1.401 V. Whch s the postve battery termnal? Note that we have used the ground symbol ( ) for the reference node on the DVM. Often t s labeled C for common. Lecture 2, Slde 13 Another Example Fnd v ab, v ca, v cb a 2 V c 1 V v cd b v bd d Note that the labelng conventon has nothng to do wth whether or not v s postve or negatve. Lecture 2, Slde 14 7
Power If an element s absorbng power (.e. f p > 0), postve charge s flowng from hgher potental to lower potental. p = v f the passve sgn conventon s used: _ v or v _ How can a crcut element absorb power? By convertng electrcal energy nto heat (resstors n toasters), lght (lght bulbs), or acoustc energy (speakers); by storng energy (chargng a battery). Lecture 2, Slde 15 Power Calculaton Example Fnd the power absorbed by each element: Conservaton of energy total power delvered equals total power absorbed Asde: For electroncs these are unrealstcally large currents mllamperes or smaller s more typcal v (W) p (W) 918-810 -12-400 - 224 1116 Lecture 2, Slde 16 8
Crcut Elements 5 deal basc crcut elements: voltage source current source resstor nductor capactor actve elements, capable of generatng electrc energy passve elements, ncapable of generatng electrc energy Many practcal systems can be modeled wth just sources and resstors The basc analytcal technques for solvng crcuts wth nductors and capactors are smlar to those for resstve crcuts Lecture 2, Slde 17 Electrcal Sources An electrcal source s a devce that s capable of convertng non-electrc energy to electrc energy and vce versa. Examples: battery: chemcal electrc dynamo (generator/motor): mechancal electrc (Ex. gasolne-powered generator, Bonnevlle dam) Electrcal sources can ether delver or absorb power Lecture 2, Slde 18 9
Ideal Voltage Source Crcut element that mantans a prescrbed voltage across ts termnals, regardless of the current flowng n those termnals. Voltage s known, but current s determned by the crcut to whch the source s connected. The voltage can be ether ndependent or dependent on a voltage or current elsewhere n the crcut, and can be constant or tme-varyng. Devce symbols: v s _ v s =µ v x _ v s =ρ x _ ndependent voltage-controlled current-controlled Lecture 2, Slde 19 Ideal Current Source Crcut element that mantans a prescrbed current through ts termnals, regardless of the voltage across those termnals. Current s known, but voltage s determned by the crcut to whch the source s connected. The current can be ether ndependent or dependent on a voltage or current elsewhere n the crcut, and can be constant or tme-varyng. Devce symbols: s s =α v x s =β x ndependent voltage-controlled current-controlled Lecture 2, Slde 20 10
Electrcal Resstance Resstance: the rato of voltage drop and current. The crcut element used to model ths behavor s the resstor. Crcut symbol: Unts: Volts per Ampere ohms (Ω) The current flowng n the resstor s proportonal to the voltage across the resstor: Georg Smon Ohm v = R (Ohm s Law) 1789-1854 where v = voltage (V), = current (A), and R = resstance (Ω) R Lecture 2, Slde 21 Electrcal Conductance Conductance s the recprocal of resstance. Symbol: G Unts: semens (S) or mhos ( ) Example: Consder an 8 Ω resstor. What s ts conductance? Ω Werner von Semens 1816-1892 Lecture 2, Slde 22 11
Example: Power Absorbed by a Resstor p = v = ( R ) = 2 R p = v = v ( v/r ) = v 2 /R Note that p > 0 always, for a resstor a resstor dsspates electrc energy Example: a) Calculate the voltage v g and current a. b) Determne the power dsspated n the 80Ω resstor. Lecture 2, Slde 23 Short Crcut and Open Crcut Short crcut R = 0 no voltage dfference exsts all ponts on the wre are at the same potental. Current can flow, as determned by the crcut Open crcut R = no current flows Voltage dfference can exst, as determned by the crcut Lecture 2, Slde 24 12
I-V Characterstc of Ideal Voltage Source a V ab _ b _ v s =0 V s >0 v 1. Plot the I-V characterstc for v s > 0. For what values of does the source absorb power? For what values of does the source release power? 2. Repeat V s >0 (1) <0 for release v s < 0. power; >0 absorb power 3. What s the I-V characterstc for an deal wre? Lecture 2, Slde 25 I-V Characterstc of Ideal Current Source v _ s v 1. Plot the I-V characterstc for s > 0. For what values of v does the source absorb power? For what values of v does the source release power? V>0 absorb power; V<0 release power Lecture 2, Slde 26 13
a v _ b I-V Characterstc of Ideal Resstor R 1. Plot the I-V characterstc for R = 1 kω. What s the slope? v a a V ab _ R V ab R b b Lecture 2, Slde 27 Summary Current = rate of charge flow = dq/dt Voltage = energy per unt charge created by charge separaton Power = energy per unt tme ( n to v => pos) Ideal Basc Crcut Elements two-termnal component that cannot be sub-dvded descrbed mathematcally n terms of ts termnal voltage and current An deal voltage source mantans a prescrbed voltage regardless of the current n the devce. An deal current source mantans a prescrbed current regardless of the voltage across the devce. A resstor constrans ts voltage and current to be proportonal to each other: v = R (Ohm s law) Lecture 2, Slde 28 14